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Nonlinear Differential Inclusions of Semimonotone and Condensing Type in Hilbert Spaces

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Nonlinear Differential Inclusions of Semimonotone and Condensing Type in Hilbert Spaces Bull. Korean Math. Soc. 52 (2015), No. 2, pp. 421–438 http://dx.doi.org/10.4134/BKMS.2015.52.2.421 NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE AND CONDENSING TYPE IN HILBERT SPACES Hossein Abedi and Ruhollah Jahanipur Abstract. In this paper, we study the existence of classical and gen- eralized solutions for nonlinear differential inclusions x′(t) ∈ F (t, x(t)) in Hilbert spaces in which the multifunction F on the right-hand side is hemicontinuous and satisfies the semimonotone condition or is condens- ing. Our existence results are obtained via the selection and fixed point methods by reducing the problem to an ordinary differential equation. We first prove the existence theorem in finite dimensional spaces and then we generalize the results to the infinite dimensional separable Hilbert spaces. Then we apply the results to prove the existence of the mild solution for semilinear evolution inclusions. At last, we give an example to illustrate the results obtained in the paper. 1. Introduction Our aim is to study the nonlinear differential inclusion of first order x′(t) ∈ F (t, x(t)), (1.1) x(0) = x0, in a Hilbert spaces in which F may be condensing or semimonotone set-valued map. An approach to investigating the existence of solution for a differential inclusion is to reduce it to a problem for ordinary differential equations. In this setting, we are interested to know under what conditions the solution of the corresponding ordinary differential equation belongs to the right side of the given differential inclusion; in other words, under what conditions a continuous function such as f(·, ·) exists so that f(t, x) ∈ F (t, x) for every (t, x) and x′(t)= f(t, x(t)). Thus, fixed point theorems and selection theorems to some of which we point out in Section 2, are the main tools in this method. The corresponding ′ ordinary differential equation x (t) = f(t, x(t)) with x(0) = x0, in case that f : R × Rn → Rn is continuous and Lipschitz (or Locally Lipschitz) with Received September 28, 2013. 2010 Mathematics Subject Classification. Primary 34A60; Secondary 49K24, 47H05. Key words and phrases. differential inclusions, set-valued integral, semimonotone and hemicontinuous multifunctions, condensing multifunctions. c 2015 Korean Mathematical Society 421 422 HOSSEINABEDIANDRUHOLLAHJAHANIPUR respect to the second variable, according to the Picard-Lindelof theorem [1], has a unique classical solution (local solution). However, it is well-known that continuity condition alone does not suffice for the problem to have even a local solution in infinite dimensional spaces. In a more general case, the function f is considered to satisfy Caratheodory condition and therefore, solutions of Caratheodory type are obtained. To study the semilinear and nonlinear Cauchy differential equations in Hilbert spaces, we refer to [9, 10, 19, 20, 22]. For differential inclusions, the conditions we impose on set-valued map F are usually a combination of two types: First, regularity of the map F such as var- ious kinds of continuity, semicontinuity and monotonicity condition. Second, geometrical conditions such as compactness, connectedness and convexity of the values of F . Existence of solution for differential inclusions have been studied by many authors in the past half century with different application-directed mo- tivations, including the issues of control and optimization, dynamical systems and even biological sciences [11, 12, 15, 16, 21]. In the earlier works, set-valued maps have been usually considered with convex values [6, 7]. Next, in the case that the images are nonconvex, the existence theorems were also studied by some authors. For example, Bressan [7] proved the existence theorems when the values of set-valued map are completely disconnected subsets of a finite di- mensional space. Also, in [2, 3] differential inclusion with convex and nonconvex values on infinite dimensional Banach spaces have systematically been studied. In the case that set-valued map F is nonexpansive or Lipschitz in Hausdorff metric topology, several inequalities such as the Gronwall inequality give us a lot of useful information about properties of the set of trajectories of solutions [2, 7, 21, 24]. However, nonexpansive and Lipschitz conditions are very strong and are not of practical importance. For example, in optimal control problems, set-valued functions are often defined as F (t, x)= {f(t,x,u): u ∈ U}, where U is a metric space and the single-valued function f is defined on [0,T ] × Rn × U into Rn . Here F need not be Lipschitz even if the function f is. Thus, replacing Lipschitz condition with a weaker one, would be very valuable. When the set-valued map A = −F is maximal monotone on an infinite di- −1 mensional Hilbert space, the resolvent map Jλ = (I +λA) and the Yosida ap- 1 proximation Aλ = λ (I −Jλ) of A are single-valued. Also, limλ→0 Aλ(x) ∈ A(x) for every x ∈ D(Jλ). In this case, like nonlinear differential equations with maximal monotone condition [23], first, a set of approximate solutions xλ ′ are obtained for differential equations xλ(t) = Aλ(xλ(t)). The images of these solutions under Aλ constitute a weakly compact subset of the space ∞ ∞ L (0,T ; H) and finally a Cauchy subsequence {xλn }n=1 is extracted such that ′ ′ 2 xλn converges weakly to x in L (0,T ; H) and xλn converges strongly to x in L2(0,T ; H). Since A is semiclosed, this subsequence is convergent to the unique solution x of (1.1). In this paper, we establish the existence and uniqueness of the classical and generalized solution for monotone-type differential inclusion (1.1). Our NONLINEAR DIFFERENTIAL INCLUSIONS OF SEMIMONOTONE TYPE 423 novelty is mainly in the conditions we will impose on the nonlinearity F and in the method we will use to prove the existence result which is, in fact, based upon continuous and measurable selection theorems and Kakutani’s fixed point theorem. The paper is organized as follows. In Section 2, we provide some definitions and preliminaries that are required in the next sections. We have allocated Sections 3 and 4 to the existence of generalized solution for problem (1.1). In these sections, we aim to present our main theorems and results when the multifunction F is semimonotone hemicontinuous or condensing. Section 5 includes an application of the results of the previous sections to the existence of the mild solution for semilinear differential inclusions. Finally, we give an example. 2. Definitions and preliminaries Let X be a Banach space. We assume that P (X) is the family of all non-empty subsets of X and Pc(X) (resp. Pc,v(X), Pk,v(X) and Pb,c,v(X)) is the family of all non-empty closed (resp. closed convex, compact convex and bounded closed convex) subsets of X. A set-valued map F : X → P (Y ) where Y is another Banach space, is called upper semicontinuous (u.s.c) if for every open subset U of Y , the set {x ∈ X : F (x) ⊆ U} is open in X. By Proposi- tion 1.2.5 of [18], upper semicontinuity is equivalent to the following: for each ∞ sequence {xn}n=1 in X such that xn → x as n → ∞ and for every ε> 0, there exists a positive integer N such that F (xn) ⊆ F (x)+ εB, ∀n ≥ N, where B is the open unit ball in Y . The map F is called lower semicontinuous (l.s.c) if for every open subset U of Y , the set {x ∈ X : F (x) ∩ U 6= ∅} is open in X. By Proposition 1.2.6 of [18], lower semicontinuity is equivalent to the ∞ following: for each sequence {xn}n=1 in X which converges to x and for each ∞ y ∈ F (x), there exists a sequence {yn}n=1 in Y such that yn ∈ F (xn) for all n and yn → y as n → ∞. Moreover, the set-valued map F is called closed when its graph {(x, y): y ∈ F (x)} is closed in the product space X × Y and it is called compact if F (X) is a compact subset of Y . If for each x ∈ X there exists a neighborhood Ux of x such that F (Ux) is relatively compact in Y , then F is called locally compact. In this paper, we apply Proposition 1.2.23 of [18] to prove upper semicontinuity of set-valued maps: Proposition 2.1. Let F : X → Pc(Y ) be closed and locally compact. Then F is u.s.c. Suppose that T > 0 and I = [0,T ] is an interval on the real line equipped with the σ-field of Lebesgue measurable sets. The set-valued map F : I → P (X) is called measurable if for every open subset U of X, the set {t ∈ I : F (t) ∩ U 6= ∅} is measurable in I. To say that F is measurable is the same as 424 HOSSEINABEDIANDRUHOLLAHJAHANIPUR for every x ∈ X, the map y : I → R defined by y(t)= d(x, F (t)) = inf{d(x, y): y ∈ F (t)} is measurable [18]. A single-valued function f : X → Y is called a selection for the multifunction F : X → P (Y ) provided that f(x) ∈ F (x) for all x ∈ X. According to the Michael selection theorem [18], if X is a metric and Y is a Banach space, then any l.s.c set-valued map F : X → Pc,v(Y ) has a continuous selection. Moreover, ∞ if X is separable, then there exists a sequence of continuous selections {fn}n=1 such that for every x ∈ X, F (x) = {fn(x)}. One may also be interested in measurable selections as we do in this paper.
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